THE WIENER CRITERION FOR FULLY NONLINEAR ELLIPTIC EQUATIONS

We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which characterizes a regular boundary point via potential theory. Our approach utilizes the asymptotic behavior of homogeneous solutions, together with Harnack inequality and the comparison principle.


Introduction
Let whenever f ∈ C(∂Ω). One simple characterization of a regular boundary point is to find a barrier function; see Section 2 for the precise definition. As a consequence, by constructing proper barrier functions, geometric criteria on ∂Ω such as an exterior sphere condition or an exterior cone condition have been invoked to guarantee the boundary continuity at x 0 ∈ ∂Ω for a variety of elliptic operators.
On the other hand, Wiener [40] developed an alternative criterion for a regular boundary point, based on potential theory. Namely, for the Laplacian operator (M = ∆), x 0 ∈ ∂Ω is regular if and only if the Wiener integral diverges, i.e.
where cap 2 (K, Ω) is defined by the variational capacity of the Laplacian operator. Surprisingly, the Wiener criterion becomes both a sufficient and necessary condition for the regularity of a boundary point. Here the notion of capacity is used to measure the 'size' of sets in view of given differential equations. Roughly speaking, x 0 ∈ ∂Ω is regular if and only if Ω c is 'thick' enough at x 0 in the potential theoretic sense. Both linear and nonlinear potential theory have been extensively studied in literature; see [5,12,13,25,31,39] and references therein. Since the main ingredient of potential theory comes from the integration by parts, the theory and corresponding Wiener criterion have been developed mostly for operators in divergence form. Littman, Stampacchia and Weinburger [30] demonstrated the coincidence between the regular points for uniformly elliptic operators M = −D j (a ij D i ), where a ij is bounded and measurable, and for the Laplacian operator. For the p-Laplacian operator (M = ∆ p , p > 1), Maz'ya [32] verified the sufficiency of the p-Wiener criterion, i.e. x 0 ∈ ∂Ω is regular for ∆ p ifˆ1 For the converse direction, Lindqvist and Martio [29] proved the necessity of the Wiener criterion under the assumption p > n − 1. Later, Kilpeläinen and Malý [20] extended this result to any p > 1, via the Wolff potential estimate. For the other available results on the Wiener criterion, we refer to [1] for p(x)-Laplacian operators and [27] for operators with Orlicz growth. Note that all of these results consider elliptic operators in divergence form. For elliptic operators in non-divergence form, relatively small amounts of results for the Wiener criterion are known. While the equivalence was obtained for M = D j (a ij D i ) with merely measurable coefficients in [30], Miller [35,36] discovered the non-equivalence with respect to M = a ij D ij u, even if the coefficients a ij are continuous. More precisely, he presented examples of linear operators M in non-divergence form and domains Ω such that x 0 ∈ ∂Ω is regular for M, but x 0 is irregular for ∆, and vice-versa. We also refer [22,26]. On the other hand, Bauman [4] developed the Wiener test for M = a ij D ij u with continuous coefficients a ij . He proved that x 0 ∈ ∂Ω is regular if and only if Here g is the normalized Green function and e is a unit vector in R n .
The goal of this paper is to establish the Wiener criterion for fully nonlinear elliptic operators, by implementing potential theoretic tools. To illustrate the issues, we consider an Issacs operator, i.e. an operator F with the following two properties: (F1) F is uniformly elliptic: there exist positive constants 0 < λ ≤ Λ such that for any M ∈ S n , Then it is obvious that F also satisfies (F1) and (F2). One important property that F satisfying (F1) and (F2) possesses is the existence of a homogeneous solution V : Lemma 1.1 (A homogeneous solution; [3,7]). There exists a non-constant solution of F (D 2 u) = 0 in R n \ {0} that is bounded below in B 1 and bounded above in R n \ B 1 . Moreover, the set of all such solutions is of the form {aV + b | a > 0, b ∈ R}, where V ∈ C 1,γ loc (R n \ {0}) can be chosen to satisfy one of the following homogeneity relations: for all t > 0 V (x) = V (tx) + log t in R n \ {0} where α * = 0, or V (x) = t α * V (tx), α * V > 0 in R n \ {0}, for some number α * ∈ (−1, ∞)\{0} that depends only on F and n. We call the number α * = α * (F ) the scaling exponent of F . Now we are ready to state our first main theorem, namely, the sufficiency of the Wiener criterion: Theorem 1.2 (The sufficiency of the Wiener crietrion). If then the boundary point x 0 ∈ ∂Ω is (F -)regular.
We remark that the Wiener integral is again defined in terms of a capacity, but the definition of a F -capacity is quite different from the variational capacity for the Laplacian case; see Section 3 for details. Furthermore, as a corollary of Theorem 1.2, we will derive the quantitative estimate for a modulus of continuity at a regular boundary point (Lemma 4.7), and suggest another geometric condition, called an exterior corkscrew condition (Corollary 4.9).
Our second main theorem is concerned with the necessity of the Wiener criterion. We propose a partial result on the necessary condition, i.e. exploiting the additional structure of F , we show that the Wiener integral at x 0 ∈ ∂Ω must diverge whenever x 0 is a regular boundary point. Theorem 1.3 (The necessity of the Wiener criterion). Suppose that F is concave and α * (F ) < 1. If a boundary point x 0 ∈ ∂Ω is regular, then Note that the assumption α * (F ) < 1 in the fully nonlinear case corresponds to the assumption p > n − 1 in the p-Laplacian case, [29]. The underlying idea for both cases is to utilize the nonzero capacity of a line segment (or a set of Hausdorff dimension 1). Further comments on this assumption can be found in Section 5.
In this paper, the main difficulty arises from the inherent lack of divergence structure; we cannot define a variational capacity by means of an energy minimizer, and moreover, we cannot employ integral estimates involving Sobolev inequality and Poincaré inequality. Instead, we will develop potential theory with non-divergence structure by the construction of appropriate barrier functions using the homogeneous solution, and by the application of the comparison principle and Harnack inequality. In short, our strategy is to capture the local boundary behavior of the upper Perron solution H f in terms of newly defined capacity cap F (K, B) and the capacity potential (or the balayage)R 1 K (B), using prescribed tools. Heuristically, the non-variational capacity measures the 'height' of the F -solution with the boundary value 0 on ∂B and 1 on ∂K, while the variational capacity measures the 'energy' of such function. We emphasize that although our notion of capacity does not satisfy the subadditive property in general, it was still able to recover certain properties of the variational capacity.
Finally, we would like to point out that the dual operator F is different from F , for general F . Thus, even though u is an F -supersolution, we cannot guarantee −u is an F -subsolution. Moreover, a similar feature is found in the growth rate of the homogeneous solution for F ; two growth rates of an upward-pointing homogeneous solution and a downward-pointing one can be different. This phenomenon naturally leads us (i) to describe the local behavior of both the upper Perron solution H f and the lower Perron solution H f for regularity at x 0 ∈ ∂Ω; (ii) to construct two (upper/lower) barrier functions when characterizing a regular boundary point; (iii) to display two different Wiener integrals in our main theorem, which differ from the previous results that appeared in [4,20,40].
Outline. This paper is organized as follows. In Section 2, we summarize the terminology and preliminary results for our main theorems. In short, we introduce F -superharmonic functions and Poisson modification and then perform Perron's method. In Section 3, we first define a balayage and a capacity for uniformly elliptic operators in non-divergence form. Then we prove several capacitary estimates by constructing auxiliary functions and provide the characterization of a regular boundary point via balayage. Section 4 consists of potential theoretic estimates for the capacity potential. Then we prove the sufficiency of the Wiener criterion and several corollaries. Finally, Section 5 is devoted to the proof of the (partial) necessity of the Wiener criterion.
2. Perron's method 2.1. F -supersolutions and F -superharmonic functions. In this subsection, we only require the condition (F1) for an operator F . To illustrate Perron's method precisely, we start with two different notions of solutions for a uniformly elliptic operator F : F -solutions and F -harmonic functions. Indeed, we will prove that these two notions coincide.
in Ω, when the following condition holds: if x 0 ∈ Ω, ϕ ∈ C 2 (Ω) and u − ϕ has a local minimum at x 0 , then [resp. if u − ϕ has a local maximum at x 0 , then F (D 2 ϕ(x 0 )) ≥ 0.] We say that u ∈ C(Ω) a (viscosity) F -solution if u is both an F -subsolution and an Fsupersolution. Proof. We argue by contradiction: suppose that Then for any ϕ ∈ C 2 (Ω), it follows that u − ϕ has a local minimum at x 0 and so we can test this function. Therefore, which is impossible.
Assume that u k converges uniformly in every compact set of Ω to u. Then u is an F -solution in Ω. (ii) (Compactness) Suppose that {u k } k≥1 ⊂ C(Ω) is a locally uniformly bounded sequence of F -solutions in Ω. Then it has a subsequence that converges locally uniformly in Ω to an F -solution.
Theorem 2.4 (Harnack convergence theorem). Let {u k } k≥1 ⊂ C(Ω) be an increasing sequence of F -solutions in Ω. Then the function u = lim k→∞ u k is either an F -solution or identically +∞ in Ω.
Proof. If u(x) < ∞ for some x ∈ Ω, it follows from Harnack inequality that u is locally bounded in Ω. The interior C α -estimate yields that the sequence u k is equicontinuous in every compact subset of Ω. Thus, applying Arzela-Ascoli theorem and Theorem 2.3 (i), we finish the proof.
We demonstrate two essential tools for Perron's method, namely, the comparison principle and the solvability of the Dirichlet problem in a ball. Theorem 2.5 (Comparison principle for F -super/subsolutions, [17,18]). Let Ω be a bounded open subset of R n . Let v ∈ USC(Ω) [resp.u ∈ LSC(Ω)] be an F -subsolution [resp. F -supersolution] in Ω and v ≤ u on ∂Ω. Then v ≤ u in Ω.
In the previous theorem, USC(Ω) denotes the set of all upper semi-continuous functions from Ω to R. Moreover, note that for a lower semi-continuous function f , there exists an increasing sequence of continuous functions {f n } such that f n → f pointwise as n → ∞.
Theorem 2.6 (The solvability of the Dirichlet problem). Let Ω satisfy a uniform exterior cone condition and f ∈ C(∂Ω). Then there exists a unique F -solution u ∈ C(Ω) of the Dirichlet problem Proof. The existence depends on the construction of global barriers achieving given boundary data and the standard Perron's method; see [9,33] and [8,15]. Then the uniqueness comes from the comparison principle, Theorem 2.5.  are real numbers and a ≥ 0. (ii) If u and v are F -superhmaronic, then the function min{u, v} is F -superharmonic. (iii) Suppose that u i , i = 1, 2, · · · , are F -superharmonic in Ω. If the sequence u i is increasing or converges uniformly on compact subsets of Ω, then in each component of Ω, the limit function u = lim i→∞ u i is F -superharmonic unless u ≡ ∞.
Theorem 2.9 (Comparison principle for F -super/subharmonic functions). Suppose that u is Fsuperharmonic and that v is F -subharmonic in Ω. If for all x ∈ ∂Ω, then v ≤ u in Ω.
Proof. Fix ε > 0 and let Then K ε is a compact subset of Ω and so there exists an open cover D ε such that K ε ⊂ D ε ⊂ Ω where D ε is a union of finitely many balls B i , and ∂D ε ⊂ Ω \ K ε . Since u is lower semi-continuous, v is upper semi-continuous and ∂D ε is compact, we can choose a continuous function θ on ∂D ε such that v ≤ θ ≤ u + ε on ∂D ε . Moreover, since D ε satisfies a uniform exterior cone condition, there exists h ∈ C(D) which is the unique F -solution in D ε that coincides with θ on ∂D ε by applying Theorem 2.6. Now the definition of F -super/subharmonic functions yields that Hence, v ≤ u + ε in Ω and the desired result follows by letting ε → 0. Now we describe the equivalence of F -supersolution and F -superharmonic function; see also [14,21,24]. Theorem 2.10. u is an F -supersolution in Ω if and only if u is F -superharmonic in Ω.
Proof. Assume first that u is an F -supersolution in Ω. To show that u is F -superharmonic, we only need to verify the property (iii) in the definition of F -superharmonic functions. Let D ⊂⊂ Ω be an open set and take h ∈ C(D) to be an F -solution in D such that h ≤ u on ∂D. Thus, applying the comparison principle for F -super/subsolutions (Theorem 2.5) for u and h, we conclude that h ≤ u in D.
The result for F -subsolution and F -subharmonic function can be derived in the same manner and consequently, a function u is an F -solution if and only if it is F -harmonic.
is lower semi-continuous, then s is F -superharmonic in Ω.
Proof. Let G ⊂⊂ Ω be open and h ∈ C(G) be F -harmonic such that h ≤ s on ∂G. Then h ≤ u in G. In particular, since s is lower semi-continuous, for all x ∈ ∂D ∩ G. Thus, for all x ∈ ∂(D ∩ G), and Theorem 2.9 implies h ≤ v in D ∩ G. Therefore, h ≤ s in G and the lemma is proved.
Suppose that u is F -superharmonic in Ω and that B ⊂⊂ Ω is an open ball. Let Then define the Poisson modification P (u, B) of u in B to be the function for any x ∈ ∂B, we have h ≥ P (u, B) in B by the definition of u B . On the other hand, since h j (x) ≤ lim inf y→x v(y) where x ∈ ∂B and v is an admissible function for u B , we have h ≤ P (u, B) in B by applying the comparison principle, letting j → ∞ and taking the infimum over v. Therefore, Finally, if we show that P (u, B) is lower semi-continuous, then it immediately follows from the pasting lemma that P (u, B) is F -superharmonic in Ω. Indeed, it is enough to show that P (u, B) is lower semi-continuous at each point x ∈ ∂B; recall (2.2).
Then, the comparison principle yields that H f ≤ H f . Proof. This proof is based on the argument used in [19]. Fix an open ball B with B ⊂⊂ Ω. Next, choose a countable, dense subset X = {x 1 , x 2 , ...} of B and then for each j = 1, 2, ..., choose Moreover, replacing u i,j+1 by min{u i,j , u i,j+1 } if necessary, we have for each k = 1, 2..., j and each j. Now, let U i,j := P (u i,j , B) be the Poisson modification of u i,j in B. Then we observe that H f ≤ U i,j ≤ u i,j and U i,j is F -harmonic in B. By compactness (Theorem 2.3 (ii)), U i,j converges locally uniformly to F -harmonic v j in B (passing to a subsequence, if necessary). Again by compactness, v j converges locally uniformly to F -harmonic h in B.
By the construction of h, it follows immediately that Hence, H f = h is F -harmonic in Ω and a similar argument for H f completes the proof.
We emphasize that although we proved that F (D 2 H f ) = 0 in Ω, we cannot guarantee that H f enjoys the boundary condition of the Dirichlet problem, H f = f on ∂Ω. To investigate the boundary behavior of the Perron solutions and ensure the solvability of the Dirichlet problem, we need to introduce further concepts, namely, a regular point and a barrier function.
. An open and bounded set Ω is called regular if each x 0 ∈ ∂Ω is a regular boundary point.
and so in this case, we can equivalently call whenever f ∈ C(∂Ω). Nevertheless, for the general fully nonlinear operator F , we do not have this property. Therefore, it seems that we have to require both conditions simultaneously, when we define a regular point for F . To the best of our knowledge, it is unknown whether the two conditions in the definition are redundant. One possible approach to show that only one condition is essential is to prove that f is resolutive whenever f is continuous on ∂Ω; see Definition 2.17 for the definition of resolutivity.
Before we define a barrier function, which characterizes a regular boundary point, we shortly deal with the resolutivity of boundary data: (i) If f = c on ∂Ω, then f is resolutive and Note that the resolutivity of f does not imply for x ∈ ∂Ω. However, the converse is true in some sense: Let Ω be an open and bounded subset of R n and f be a bounded function on ∂Ω.
we conclude that f is resolutive. An analogous argument works for the F -subharmonic case.

2.3.
Characterization of a regular point.
Observe that the maximum principle indicates that an upper barrier w + is positive in Ω and a lower barrier w − is negative in Ω. Moreover, under the condition (F2), cw + is still an upper barrier for any constant c > 0 and an upper barrier w + . See also [38]. Now we can deduce that a regular boundary point is characterized by the existence of upper and lower barriers.
Here we used that x → lim inf Ω∋y→x w + (y) is lower semi-continuous on ∂Ω.
An analogous argument leads to .
i.e. x 0 is a regular boundary point.
and so w + is a desired upper barrier. The existence of a lower barrier is guaranteed by considering Indeed, the barrier characterization is a local property: Proof. By Theorem 2.22, there exist an upper barrier w + and a lower barrier w − with respect to Ω at x 0 . Then w + | G and w − | G become the desired barriers with respect to G at x 0 . Again by Theorem 2.22, x 0 is regular with respect to G. Proof. By Lemma 2.23, one direction is immediate. For the opposite direction, suppose that x 0 is regular with respect to B ∩ Ω. Then there exist an upper barrier w + and a lower barrier w − with respect to B ∩ Ω. If we let m := min ∂B∩Ω w + > 0 (the minimum exists because w + is lower semi-continuous), then the pasting lemma, Lemma 2.11, shows that is F -superharmonic in Ω. One can easily verify that s + is an upper barrier with respect to Ω at x 0 . Similarly, a lower barrier s − can be constructed.
The barrier characterization leads to another useful corollary, which enables us to write x 0 is regular instead of F -regular, without ambiguity.
Proof. Suppose that x 0 is F -regular. By Theorem 2.22, there exists an upper barrier w + F and a lower barrier w − F . If we let w + F := −w − F and w − F := −w + F , then w + F and w − F become an upper barrier and a lower barrier for F , respectively. Therefore, again by Theorem 2.22, x 0 is F -regular. Now we present one sufficient condition that guarantees a regular boundary point, namely the exterior cone condition. In Section 4, we suggest another sufficient condition, namely the Wiener criterion, which contains this exterior cone condition as a special case.  Proof. Since polyhedra and balls satisfy the uniform exterior cone condition, the first assertion follows from Theorem 2.26. For the second assertion, exhaust Ω by domains D 1 ⊂⊂ D 2 ⊂⊂ · · · ⊂⊂ Ω. Then, since D j is compact, there exists a finite union of open cubes Q ji (⊂ D j+1 ) that covers D j . Letting P j := i int Q ji which is a polyhedron by the construction, we obtain the desired exhaustion.

Balayage and capacity
3.1. Balayage and capacity potential. We define the lower semi-continuous regularizationû of any function u : Lemma 3.1. Suppose that F is a family of F -superharmonic functions in Ω, locally uniformly bounded below. Then the lower semi-continuous regularization s of inf F , Proof. Since F is locally uniformly bounded below, s is lower semi-continuous. Fix an open D ⊂⊂ Ω and let h ∈ C(D) be an F -harmonic function satisfying h ≤ s on ∂D. Then h ≤ u in D whenever u ∈ F . It follows from the continuity of h that h ≤ s in D. Then the function is called the reduced function and its lower semi-continuous regularization The functionR u E is called the balayage of u relative to E. (iii) In particular, we call the functionR 1 E the (F -)capacity potential of E in Ω. Remark 3.3. For an operator in divergence form, there exists an alternative method to define the capacity potential. For simplicity, suppose that the operator is given by the p-Laplacian. Let Ω be bounded and K ⊂ Ω be a compact set. For ψ ∈ C ∞ 0 (Ω) with ψ ≡ 1 on K, the p-harmonic function u in Ω \ K with u − ψ ∈ W 1,p 0 (Ω \ K) is called the capacity potential of K in Ω and denoted by R(K, Ω). Here note that R(K, Ω) is independent of the particular choice of ψ and the existence of the capacity potential is guaranteed by the variational method. Indeed, both definitions of capacity potentials coincide; see [12,Chapter 9] for details.
Proof. Observe first that if v 1 and v 2 are in Φ u E , then so is min{v 1 , v 2 }. Hence, the family Φ u E is downward directed and we may invoke Choquet's topological lemma (see Lemma 8.3. in [12]): there is a decreasing sequence of Next, we choose a ball B ⊂⊂ Ω \ E and consider a Poisson modification which implies thatR u E =v =ŝ. Moreover, since s is F -harmonic in B (Harnack convergence theorem, Theorem 2.4), we know thatŝ = s. Therefore, we conclude that the balayageR u E is F -harmonic in Ω \ E. The second assertion of the lemma is rather immediate since Lemma 3.5. Let K be a compact subset of Ω and consider R 1 , (iv) It immediately follows from Lemma 3.4 and part (ii).
The following theorem shows that the capacity potential can be understood as the upper Perron solution: Theorem 3.6. Suppose that K is a compact subset of a bounded, open set Ω and that u =R 1 K (Ω) is the capacity potential of K in Ω. Moreover, let f be a function such that is F -superharmonic in Ω by pasting lemma, Lemma 2.11. Obviously, s ∈ Φ 1 K and so R 1 3.2. Capacity. In general, for an operator in divergence form, we consider a variational capacity, which comes from minimizing the energy among admissible functions. On the other hand, for an operator in non-divergence form, we cannot consider the corresponding energy, and so we require an alternative approach to attain a proper notion of capacity. Our definition of a capacity is in the same context with Bauman [4] (for linear operators in non-divergence form) and Labutin [24] (for the Pucci extremal operators). Finally, considering Harnack inequality forR 1 K (B) on the sphere ∂B 3r/2 (x 0 ), we notice that capacities defined for different choices of y 0 ∈ ∂B 3r/2 (x 0 ) are comparable.  (K, B), where K is a compact subset of B ′ = B 7/5r (x 0 ), enjoys the following properties: (iv) (Subadditivity) We further suppose that F is convex. If K 1 and K 2 are compact subsets of B ′ , then Proof. (i) Recalling Lemma 3.5, we have 0 ≤ cap(K, B) ≤ 1.
Since F is convex, we can apply [6, Theorem 5.8] to obtain 1 2 Putting the infimum on this inequality and evaluating at y 0 , we conclude that We would like to remove the restriction of compact sets when defining a capacity.  (ii) Define a sequence of compact sets {K j } ∞ j=1 by K j := {x ∈ R n : dist(x, K) ≤ 1/j}, and a sequence of open sets {U j } ∞ j=1 by U j := {x ∈ R n : dist(x, K) < 1/j}. We may assume K 1 ⊂ B ′ . Then we have Applying Roughly speaking, we have the following correspondance: the variational capacity ←→ divergence operator, the height capacity ←→ non-divergence operator.
In the following lemma, we explain why the definition of height capacity is reasonable in some sense. In other words, we claim that for the Laplacian operator ∆, two definitions of capacity are comparable. Proof. We may assume x 0 = 0. We denote by u the capacity potential with respect to K in B. Note that u is harmonic in B \ K.
We begin with the variational capacity: Here we applied the divergence theorem and used the behavior of u on the boundary. On the other hand, recalling the definition of height capacity, we have cap ∆,height (K, B) = u(y 0 ).
By Harnack inequality, there exist constants c 1 , c 2 > 0 which only depend on n such that Thus, if we set m − := min ∂B 3r/2 u and m + := max ∂B 3r/2 u, then we have Moreover, we consider two barriers h ± which solve the Dirichlet problem in B 2r \ B 3r/2 : Indeed, using the homogeneous solution V (x) = |x| 2−n , one can compute h ± explicitly: Then the comparison principle between u and h ± leads to Therefore, we conclude that c 1 (n)r n−2 cap ∆,height (K, B) ≤ cap ∆,var (K, B) ≤ c 2 (n)r n−2 cap ∆,height (K, B).
Next, we estimate the capacity of a ball B ρ with respect to the larger ball B 2r . Indeed, the capacity of a ball can capture the growth rate of the homogeneous solution V of F . r (x 0 ) and y 0 = x 0 + 3 2 re 1 . Then for any 0 < ρ < 7 5 r, there exists a constant c = c(n, λ, Λ) > 0 which is independent of r and ρ such that Proof. We may assume x 0 = 0. Applying the argument after the definition of a capacity, we have where the boundary data f is given by

Moreover, since a ball is a regular domain, we can write
Note that H f (B 2r \ B ρ ) is continuous upto the boundary. We now split three cases according to the sign of α * (F ).
(i) (α * > 0) In this case, for the homogeneous solution V (x) = |x| −α * V x |x| , denote V + := max |x|=1 V (x) and V − := min |x|=1 V (x) and choose two points x + , x − with |x + | = 1 = |x − | so that Then we have Thus, the comparison principle yields that Finally, applying Harnack inequality for v on ∂B 3r/2 , there exists a constant c 1 > 0 which is independent of r > 0 such that Therefore, we have the desired upper bound: Similarly, we derive the lower bound: (ii) (α * < 0) For simplicity, we assume that the upward-pointing homogeneous solution is given by Then we can explicitly write the capacity potential:

For general V , we can compute by a similar argument as in part (i). For example, if
(iii) (α * = 0) Again for simplicity, we may assume the upward-pointing homogeneous solution is given by Similarly, we can explicitly write the capacity potential: v(x) = log(2r) − log |x| log(2r) − log ρ . Thus, For general V , we can compute by a similar argument as in part (i).
We can observe that the capacity of a single point is determined according to the sign of the scaling exponent α * (F ). In fact, one can expect the results of the following lemma taking ρ → 0 + in the capacitary estimate, Lemma 3.11. Lemma 3.12. For z 0 ∈ R n , choose a ball B = B 2r (x 0 ) so that z 0 ∈ B ′ = B 7r/5 (x 0 ).
(i) If α * (F ) ≥ 0, then be the homogeneous solution of F . Then for m := min x∈∂B V (x − z 0 ) and any ε > 0, we have due to the minimum principle and lim x→z0 V (x − z 0 ) = ∞. Thus, Since ε > 0 is arbitrary, we finish the first part of proof.
Since sup ∂B u = 0 and V is a homogeneous function, we have sup ∂B 7/5r u > 0. On the other hand, recalling Theorem 3.6, where the boundary data f is given by Then u ∈ L f and so H f (Ω \ {z 0 }) ≥ u. Therefore, we conclude that According to Lemma 3.12 (i), we immediately notice that every single point is of F -capacity zero if α * (F ) ≥ 0. Indeed, we are going to show that: to check whether a compact set K is of capacity zero or not, it is enough to test with respect to one ball B (Corollary 3.15). For this purpose, we require the following version of a capacitary estimate, called "comparable lemma". Lemma 3.14 (Comparable lemma). If K ⊂ B ′ = B 7r/5 and 0 < r ≤ s ≤ 2r, then there exists a universal constant c > 0 such that Proof. We may assume x 0 = 0. We claim that for 0 < r ≤ s ≤ 21 20 r, we have Indeed, we may iterate this inequality finitely many times to conclude the desired inequality for 0 < r ≤ s ≤ 2r. Moreover, let y r = 3 2 re 1 , y s = 3 2 se 1 and denote u r : By the definition of the capacity potential, it is immediate that u r ≤ u s in B 2r . In particular, we have cap F (K, B 2r ) = u r (y r ) ≤ u s (y r ).
On the other hand, an application of Harnack inequality for u s (in a small neighborhood of B 3s/2 \ B 10s/7 ) yields that there exists a constant c > 0 which is independent of the choice of r and s such that u s (y r ) ≤ cu s (y s ) = c cap F (K, B 2s ).
Next, for the second inequality, we first assume that α * (F ) > 0 and the homogeneous solution is given by V (x) = |x| −α * (for computational simplicity) and let Then recalling Theorem 3.6, the comparison principle yields that Then it can be easily checked that the function is F -harmonic in B 2s \ B 3r/2 and by the comparison principle, w ≥ u s in B 2s \ B 3r/2 . (here again note that 7 5 s < 3 2 r.) In particular, we obtain  (ii) for any ball B 2 such that B ′ 2 ⊃ B, we have cap(K, B 2 ) = 0; (iii) K is of F -capacity zero.
Proof. (i) Apply the first inequality of Lemma 3.14 finitely many times.
(ii) Apply the second inequality of Lemma 3.14 finitely many times.
(iii) It is an immediate consequence of (i) and (ii). Now we shortly illustrate the potential theoretic meaning of capacity zero sets, at least for convex operators F . In the end, F -capacity zero sets are 'negligible' in view of the fully nonlinear operator F ; i.e. F -capacity really measures the size of given sets in a suitable way to interpret the corresponding PDE.  Note that we still have u is F -superharmonic in B 2r and u| K = ∞. Therefore, for any ε > 0, we have εu ∈ Φ 1 K (B 2r ) and soR 1 K (B 2r ) ≤ εu. Letting ε → 0 and taking x = x 0 , we notice thatR 1 K (B 2r )(x 0 ) = 0. Finally, the strong minimum principle implies that cap F K = 0. (ii) =⇒ (i): Let y 0 = x 0 + 3 2 re 1 . Then by the definition of the capacity and the capacity potential, we haveR 1 K (B 2r (x 0 ))(y 0 ) = 0. Thus, there exists a sequence of F -superharmonic functions {u j } ∞ j=1 in B 2r such that u j ≥ 0 in B 2r , u j ≥ 1 on K and u j (y 0 ) < 1/2 j .
Define v k := k j=1 u j which is lower semi-continuous and is finite in a dense subset of Ω. Furthermore, since F is convex, we have F (D 2 v k ) ≤ 0, and so v k is F -superharmonic. Since {v k } k is an increasing sequence of F -superharmonic functions, Lemma 2.8 (iii) gives that the limit function v = v k is either F -superharmonic or v ≡ ∞. The second possibility is excluded because 0 ≤ v(y 0 ) ≤ 1. Therefore, v is F -superharmonic in B 2r and v| K = ∞, which implies that K is polar.
Definition 3.18 (Removable sets). A compact set K(⊂ B 7r/5 ) is called F -removable, or simply removable, if for each function u that is F -superharmonic on B 2r \ K and is bounded below in a neighborhood of K, there exists an extension U of u which is F -superharmonic in B 2r and U = u in B 2r \ K. Lemma 3.19. Suppose that K is a compact set of capacity zero and F is convex. Then K is removable.
Proof. Let u be an F -superharmonic function in B 2r \ K and is bounded below in a neighborhood of K. Since K is of capacity zero, we haveR 1 K (B 2r )(y 0 ) = 0 and soR 1 K (B 2r ) ≡ 0 by the strong minimum principle. In particular, R 1 K (B 2r ) ≡ 0 in B 2r \ K. Now, for any z 0 ∈ B 2r \ K, following the proof of [(ii) =⇒ (i)] part in Lemma 3.17, there exists a non-negative F -superharmonic function v z0 in B 2r such that v z0 | K = ∞ and v z0 (z 0 ) < ∞. Now we consider a canonical lower semi-continuous extension U of u across K, which is defined by Then U is the lower semi-continuous regularization of the See [11] for details. Moreover, by Lemma 2.2 and Lemma 2.10, we notice that U = u in B 2r \ K and so U is F -superharmonic in B 2r \ K.
Then we claim that U + εv z0 is F -superharmonic in B 2r , for any ε > 0 and z 0 ∈ B 2r \ K. Indeed, the convexity of F immediately guarantees that U + εv z0 is F -superharmonic in B 2r \ K. On the other hand, since U + εv z0 | K = ∞, we cannot choose any test functions for U + εv z0 at points in K. In other words, for any ϕ ∈ C 2 (Ω), U + εv z0 − ϕ cannot have a local minimum at x 0 ∈ K. Thus, recalling the equivalence of F -supersolution and F -superharmonic function (Theorem 2.10), we conclude that U + εv z0 is F -superharmonic in B 2r . Now let F = {U + εv z0 } ε>0,z0∈B2r \K be a family of F -superharmonic functions in B 2r . Since u is bounded below in a neighborhood of K and v z0 is non-negative, any element in F is locally uniformly bounded below. Thus, applying Lemma 3.1, we have On the other hand, it is easy to check that Therefore, we conclude that s = U and U is a desired extension of u.
Remark 3.20. Considering the dual operator F , one can obtain analogous definitions and corresponding results when the operator is concave. For similar results concerning polar sets and removable sets, see [12] for p-Laplacian operators, [24] for Pucci extremal operators, and [23] for k-Hessian operators. See also [2,10,11] for the analysis of polar sets and removable sets in view of Riesz capacity or Hausdorff measure.

3.4.
Another characterization of a regular point. The definitions of a reduced function and a balayage depend on the choice of an operator F . In this subsection, we need to distinguish an operator and its dual operator, so we will specify the dependence by denotingR 1,F K (Ω) orR 1, F K (Ω). We now provide a key lemma for our first main theorem, the sufficiency of the Wiener criterion: Proof. For f ∈ C(∂Ω), consider the upper Perron solution H f = H f (Ω). We may assume f (x 0 ) = 0 and max ∂Ω |f | ≤ 1. For ε > 0, we can choose a ball B with center x 0 such that ∂(2B) ∩ Ω = ∅ and |f | < ε in 2B ∩ ∂Ω. Then we define On the other hand, by Theorem 3.6,R 1, F B\Ω (2B) can be considered as the upper Perron solution for the operator F . Then since a ball is regular, we have lim y→xR 1, F B\Ω (2B)(y) = 0 for all x ∈ ∂(2B). Thus, u is continuous in Ω and by the pasting lemma, u is F -superharmonic in Ω. Moreover, it can be easily checked that lim inf y→x u(y) ≥ f (x) for any x ∈ ∂Ω. Therefore, u ∈ U f and so H f ≤ u. In particular, For the converse inequality, we define Then by a similar argument, v ∈ L f and so, Consequently, since ε > 0 is arbitrary, we conclude that i.e. x 0 is regular.
Next, we exhibit a converse direction of the above lemma: i.e. a characterization of an irregular boundary point. We expect that this lemma may be employed to prove the necessity of the Wiener criterion for the general case.
Proof. Since the capacity potential u is the lower semi-continuous regularization, we have Moreover, by definition, we have u ρ ′ ≤ u ρ when 0 < ρ ′ < ρ. Thus, we can choose a sufficiently small ρ > 0 such that (3.3) holds and Ω ∩ ∂B 2ρ (x 0 ) = ∅. Now we define a smooth boundary data f on ∂( ) and f (x) = 0 on the remaining part of ∂(Ω ∩ B 2ρ (x 0 )). Then we consider the lower Perron solution H f (Ω ∩ B 2ρ (x 0 )). We claim that the following inequality holds: Recalling the comparison principle, it is enough to check the above inequality on the boundary of the domain Ω ∩ B 2ρ (x 0 ). For this purpose, let v ∈ L f (Ω ∩ B 2ρ (x 0 )) and w ∈ U g (B 2ρ (x 0 ) \ (B ρ (x 0 ) \ Ω)) where g is given by (recall Theorem 3.6) (ii) (on Ω ∩ ∂B 2ρ (x 0 )) Similarly, we obtain Now since v and w are F -subharmonic and F -superharmonic, respectively, we derive that Taking the supremum on v and the infimum on w, we conclude (3.4) which implies that lim inf Therefore, x 0 is irregular with respect to Ω ∩ B 2ρ (x 0 ). Recalling Lemma 2.24, we deduce that x 0 is irregular with respect to Ω.

A sufficient condition for the regularity of a boundary point
In this section, we prove the sufficiency of the Wiener criterion and its sequential corollaries, via the potential estimates. More precisely, we first develop quantitative estimates for the capacity potentialR 1 K (B) by employing capacitary estimates obtained in Section 3. Then we adopt the characterization of a regular boundary point in terms of the capacity potential to deduce the desired conclusion.
For simplicity, we write for the capacity density function in (4.1).

Remark 4.2.
Recalling Lemma 3.11, there exists a constant c > 0 which is independent of t > 0 such that Thus, one may write an equivalent form of (4.1): which is a similar form to the Wiener integral appearing in [20,40].
Now we can state an equivalent form of our main theorem, Theorem 1.2: If Ω c is both F -thick and F -thick at a boundary point x 0 ∈ ∂Ω, then x 0 is regular.
To prove this statement, we need several auxiliary lemmas regarding the capacity potential.
Proof. We write v γ :=R 1 Kγ (B). Then by Lemma 3.4 and the definition of a reduced function, Then for u ∈ U fγ (B \ K γ ), we have for any x ∈ ∂K γ . Since u is F -superharmonic and v/γ is F -harmonic in B \ K γ , the comparison principle leads to u ≥ v/γ in B \ K γ and so Consequently, we conclude that Since v is a non-negative F -solution in B \K, Harnack inequality yields that there exists a constant c 1 > 0 independent of r > 0 such that  We may rewrite the previous lemma aŝ Bρ(x0)\Ω (B 2ρ (x 0 )). Then for all 0 < r ≤ ρ, there exists a constant c > 0 such that for any x ∈ B r (x 0 ).
Proof. Denote B i = B 2 1−i ρ (x 0 ). Fix 0 < r ≤ ρ and let k be the integer with 2 −k ρ < r ≤ 2 1−k ρ. Then write for i = 0, 1, 2, ... v i :=R 1 Bi+1\Ω (B i ) and Since e t ≥ 1 + t, estimate (4.5) yields that Thus, denoting m 0 := inf B1 v 0 , we have Next, let D 1 := B 1 \ (B 2 ∩ Ω c ) and Then we write u 1 :=R ψ1 (B 1 ) be the balayage with respect to the ψ 1 in B 1 . It immediately follows from the definition of balayage that Again, denoting m 1 := inf B2 u 1 , we obtain a 0 )). Now iterate this step: let D i := B i \ (B i+1 ∩ Ω c ) and Denoting u i :=R ψi (B i ) and m i := inf Bi+1 u i , we have Furthermore, we claim that is given by Therefore, by the comparison principle, u ≥ u i+1 in D i+1 and so Repeating the argument above, we conclude that v 0 ≥ u 1 ≥ · · · ≥ u k in B k , which implies that Finally, the result follows fromˆρ which can be easily checked from the dyadic decomposition. Indeed, we can deduce from Lemma 3.11 and Lemma 3.14 that if t ≤ s ≤ 2t, then where the comparable constant only depends on n, λ, Λ and these results also hold for cap F (·).
Now we are ready to prove the sufficiency of the Wiener criterion, Theorem 1.2.
Proof of Theorem 1.2. Let x 0 ∈ ∂Ω, ρ > 0 and define Then applying Lemma 4.5 for both functions, we have that for all 0 < r ≤ ρ, there exist a constant c 1 , c 2 > 0 such that for any x ∈ B r (x 0 ). Letting r → 0 + , we conclude that Since ρ > 0 can be arbitrarily chosen, an application of Lemma 3.21 yields that x 0 ∈ ∂Ω is a regular boundary point. (Note that a boundary point x 0 is F -regular if and only if it is F -regular; Corollary 2.25.) On the other hand, if additional information is imposed on the boundary data f , i.e. the boundary data f has its maximum (or minimum) at x 0 ∈ ∂Ω, then we can deduce the continuity of the Perron solution at x 0 under a relaxed condition: Corollary 4.6. Suppose that f ∈ C(∂Ω) attains its maximum [resp. minimum] at x 0 ∈ ∂Ω. If Ω c is F -thick [resp. F -thick] at x 0 ∈ ∂Ω, then Proof. Similarly as in the proof of the previous theorem, this corollary is the consequence of Lemma 3.21 and Lemma 4.5.
Furthermore, if the given boundary data f ∈ C(∂Ω) is resolutive, then we are able to obtain a quantitative estimate for the modulus of continuity. If x 0 ∈ ∂Ω with f (x 0 ) = 0, then for 0 < r ≤ ρ, we have sup Furthermore, if f is resolutive, then we have the quantitative estimate: where (B 2ρ (x 0 )) be the capacity potential of B ρ \ Ω with respect to B 2ρ .
Then let w := 1 − v and write Note that since we assumed f (x 0 ) = 0, we have max ∂Ω f ≥ 0 and max ∂Ω2ρ f ≥ 0. For u ∈ L F f , u is F -subharmonic and s is F -harmonic in Ω 2ρ . Moreover, Thus, the comparison principle yields that s ≥ u in Ω 2ρ and so s ≥ H F f in Ω 2ρ . On the other hand, let By the same argument, we derive s ≥ H F −f = −H F f in ∂Ω 2ρ . An application of Lemma 4.5 for w (and w) finishes the proof. Now we present a new geometric condition for a regular boundary point, namely the exterior corkscrew condition; see also [16,28].
Definition 4.8. We say that Ω satisfies the exterior corkscrew condition at x 0 ∈ ∂Ω if there exists 0 < δ < 1/4 and R > 0 such that for any 0 < r < R, there exists y ∈ B r (x 0 ) such that B δr (y) ⊂ Ω c ∩ B r (x 0 ). Note that if Ω satisfies an exterior cone condition at x 0 ∈ ∂Ω, then Ω satisfies an exterior corkscrew condition at x 0 . Thus, the following corollary obtained from the (potential theoretic) Wiener criterion is a generalized result of Theorem 2.26. Proof. A small modification of Lemma 3.11 and its proof, we have cap(B δr (y), B 2r (x 0 )) ∼ 1, for δ ∈ (0, 1/4) and B δr (y) ⊂ B 2r (x 0 ), where the comparable constant depends only on n, λ, Λ and δ. Thus, if x 0 satisfies an exterior corkscrew condition, then we havê and so x 0 is a regular boundary point by the Wiener criterion.
Thus, choosing ρ = r 1/2 , we conclude that the Perron solution H f is Hölder continuous at x 0 .
(i) Since α * (F ) < 0, we know that a single point has non-zero capacity. More precisely, recalling the homogeneous solution for F is given by there exists a constant c = c(λ, Λ) > 0 such that Therefore, we haveˆρ In other words, Ω c is F -thick at 0. (ii) On the other hand, since α * ( F ) > 0, we know that a single point is of capacity zero. Therefore, we haveˆρ In other words, Ω c is not F -thick at 0 and we cannot apply our Wiener's criterion. (iii) Let f 1 ∈ C(∂Ω) is a boundary data given by Then clearly the function u(x) = 1 − |x| 1− λ Λ = 1 − V (x) is the solution for this Dirichlet problem. In particular, in this case, we have H f1 = H f1 (i.e. f 1 is resolutive) and lim Ω∋x→0 H f1 (x) = 1 = f 1 (0).
Therefore, we deduce that H f2 = H f2 = 0. Furthermore, it follows that which implies that 0 is an irregular boundary point for Ω.

A necessary condition for the regularity of a boundary point
In this section, we provide the necessity of the Wiener criterion, under additional structure on the operator F . Indeed, our strategy is to employ the argument made in [29] which proved the necessity of the p-Wiener criterion for p-Laplacian operator with p > n − 1. Since the assumption p > n − 1 was essentially imposed to ensure the capacity of a line segment is non-zero in [29], we begin with finding the corresponding assumptions in the fully nonlinear case.
Lemma 5.1. Suppose that F is convex and α * (F ) > s for some s > 0. Let K be a compact subset in B r (⊂ R n ) such that H s (K) < ∞, where H s is the s-dimensional Hausdorff measure. Then Proof. For any δ > 0, define where the infimum is taken over all countable covers of K by balls B i with diameter r i not exceeding δ. Then since sup δ>0 H s δ (K) = lim δ→0 H s δ (K) = H s (K) < ∞ and K is compact, for each δ ∈ (0, r), there exist finitely many open balls Now we consider the homogeneous solution V (x) = |x| −α * V x |x| of F . Here we may assume min |x|=1 V (x) = 1 by normalizing V . If we let W i (x) := r α * i V (x − x i ), then it immediately follows that W i is non-negative and F -superharmonic in R n , and W i (x) ≥ 1 on B i . Finally, we let W := N i=1 W i (≥ 0). Since F is convex, W is F -superharmonic in R n . Moreover, W ≥ 1 on N i=1 B i and in particular, W ≥ 1 on K. Therefore, W ∈ Φ 1 K (B 4r ) and so where we used (5.1) and α * > s. Letting δ → 0, we finish the proof. Now we prove the partial converse statement of Lemma 5.1. Indeed, here we only consider the compact set K is given by a line segment L, whose Hausdorff dimension is exactly 1.
Lemma 5.2. Suppose that F is concave and α * (F ) < 1. Let L = {x 0 + se : ar ≤ s ≤ br} be a line segment in B r (x 0 ), where e is an unit vector in R n and 0 < a < b < 1 are constants satisfying b − a < 1 2 . Then cap F (L, B 2r ) > 0.
Proof. Note that since L is a line segment, for any δ > 0, one can cover L by open balls . We write such cover by K δ := N (δ) i=1 B i . Recalling Lemma 3.9 and its proof, for any ε > 0, there exist a sufficiently small δ > 0 and corresponding cover K δ such that On the other hand, for simplicity, we suppose that the homogeneous solution V is given by and α * (F ) ∈ (0, 1). Note that if α * < 0, then a single point has a positive capacity (Lemma 3.12) and the result immediately follows. Other cases can be shown by similar argument as in Lemma 5.1. For each i = 1, 2, · · · , N (δ), write Here we used the condition α * < 1.
Therefore, for we have W is F -subharmonic in B \ K δ , W ≤ 0 on ∂B 2r , and W ≤ 1 on ∂ K δ .
Note that since K δ and B 2r are regular domains, the capacity potentialR 1 K δ (B 2r ) satisfies: Hence, the comparison principle yields that In particular, putting x = x 0 + 3 2 re, we conclude that |x − x i | ≤ 3r/2 − ar = 3 2 − a r, and soR Finally, by applying Harnack inequality forR 1 K δ (B 2r ) on ∂B 3r/2 , we have Since ε > 0 is arbitrary, we finish the proof.
The idea of the previous lemma can be modified to derive the 'spherical symmetrization' result: Lemma 5.3 (Spherical symmetrization). Suppose that F is concave and α * (F ) < 1. Let K be a compact subset in B r (x 0 ) such that K meets S(t) := {x ∈ R n : |x − x 0 | = t} for all t ∈ (ar, br), where 0 < a < b < 1 are constants satisfying b < 1 4 . Then there exists a constant c = c(n, F, a, b) such that cap F (K, B 2r ) ≥ c(n, F, a, b) > 0.
Therefore, by choosing c 1 = 1 c(n,F ) + 1, we arrive at a contradiction. Now we are ready to prove the necessity of the Wiener criterion, Theorem 1.3.
Next, by Lemma 2.27 and Lemma 3.9, for each i, choose a regular domain E i such that B ri \Ω ⊂ E i and b i := cap F (E i , B 2ri ) < a i + ε · 2 −i .
Next, let f ∈ C(∂Ω) be the boundary function defined by Then we have the following results for the lower Perron solution H f = H f (Ω): (i) H f ≡ 1: Choose r > 0 large enough so that Ω ⊂ B r . Moreover, set a domain Ω 0 := B r \ (B t2 ∩ Ω) and a boundary function f 0 ∈ C(∂Ω 0 ) by Then since B r is regular, we have H f0 (Ω 0 ) < 1 in B r \ B t2 . On the other hand, for any v ∈ L f (Ω) and w ∈ U f0 (Ω 0 ), one can check that v ≤ w in Ω using the comparison principle. Therefore, we conclude that H f (Ω) ≤ H f0 (Ω 0 ) and so H f (Ω) ≡ 1. Thus, the comparison principle yields that u ≤ u 3 in B t2 \ E 3 . In particular, since S(t 3 ) ⊂ A 3 , we observe that u ≤ u 3 < γ 3 on S(t 3 ).
Iterating this argument (for example, consider u − γ 3 instead of u), we conclude that which leads to (5.2). Finally, recalling the definition of u, the estimate (5.2) is equivalent to which implies that x 0 ∈ ∂Ω is an irregular boundary point.